Working with log-ratio coordinates in coda.base
Author: Marc Comas-Cufí
Date: 2023-11-25

# Working with log-ratio coordinates in `coda.base`

#### Marc Comas-Cufí

#### 2023-11-25

In this vignette we show how to define log-ratio coordinates using `coda.base` package and its function `coordinates` with parameters `X`, a composition, and `basis`, defining the independent log-contrasts for building the coordinates.

In this vignette we work with a subcomposition of the results obtained in different regions of Catalonia in 2017’s parliament elections:

    library(coda.base)
    data('parliament2017')
    X = parliament2017[,c('erc','jxcat','psc','cs')]

# Log-ratio coordinates with `coda.base`

## The additive logratio (alr) coordinates

The alr coordinates are accessible by setting the parameter `basis='alr'` or by using the building function `alr_basis()`.

If you don’t want the last part in the denominator, the easiest way to define an alr-coordinates is to set `basis='alr'`:

    H1.alr = coordinates(X, basis = 'alr')
    head(H1.alr)
    #>          alr1        alr2       alr3
    #> 1  0.23864536 0.446503630 -0.7201917
    #> 2 -0.10388120 0.216858085 -1.0473730
    #> 3  0.36723896 0.542010167 -0.5320675
    #> 4  0.53209369 0.798479995 -0.4799141
    #> 5  0.54918649 0.477309280 -0.1028807
    #> 6 -0.09742133 0.002856425 -0.6858265

It defines an alr-coordinates were the last part is used in the denominator. We can obtain the basis used to build the coordinates with function `basis()`:

    basis(H1.alr)
    #>       alr1 alr2 alr3
    #> erc      1    0    0
    #> jxcat    0    1    0
    #> psc      0    0    1
    #> cs      -1   -1   -1

The basis can be reproduced using the function `alr_basis`:

    alr_basis(dim = 4)
    #>    alr1 alr2 alr3
    #> c1    1    0    0
    #> c2    0    1    0
    #> c3    0    0    1
    #> c4   -1   -1   -1

In fact, function `alr_basis` allows to define any type of alr-like coordinate by defining the numerator and the denominator:

    B.alr = alr_basis(dim = 4, numerator = c(4,2,3), denominator = 1)
    B.alr
    #>    alr1 alr2 alr3
    #> c1   -1   -1   -1
    #> c2    0    1    0
    #> c3    0    0    1
    #> c4    1    0    0

The log-contrast matrix can be used as `basis` parameter in `coordinates()` function:

    H2.alr = coordinates(X, basis = B.alr)
    basis(H2.alr)
    #>    alr1 alr2 alr3
    #> c1   -1   -1   -1
    #> c2    0    1    0
    #> c3    0    0    1
    #> c4    1    0    0

## The centered logratio (clr) coordinates

Building centered log-ratio coordinates can be accomplished by setting parameter `basis='clr'` or

    H.clr = coordinates(X, basis = 'clr')
    head(H.clr)
    #>         clr1      clr2       clr3         clr4
    #> 1 0.24740605 0.4552643 -0.7114311  0.008760689
    #> 2 0.12971783 0.4504571 -0.8137740  0.233599031
    #> 3 0.27294355 0.4477148 -0.6263629 -0.094295406
    #> 4 0.31942879 0.5858151 -0.6925790 -0.212664904
    #> 5 0.31828271 0.2464055 -0.3337844 -0.230903777
    #> 6 0.09767651 0.1979543 -0.4907286  0.195097842

## The isometric logratio (ilr) coordinates

`coda.base` allows to define a wide variety of ilr-coordinates: principal components (pc) coordinates, specific user balances coordinates, principal balances (pb) coordinates, balanced coordinates (default’s CoDaPack’s coordinates).

The default ilr coordinate used by `coda.base` are accessible by simply calling function `coordinates` without parameters:

    H1.ilr = coordinates(X)
    head(H1.ilr)
    #>          ilr1      ilr2        ilr3
    #> 1 -0.14697799 0.8677450 -0.01011597
    #> 2 -0.22679692 0.9012991 -0.26973693
    #> 3 -0.12358191 0.8056307  0.10888296
    #> 4 -0.18836356 0.9350526  0.24556428
    #> 5  0.05082486 0.5030669  0.26662472
    #> 6 -0.07090708 0.5213690 -0.22527958

Parameter `basis` is set to `ilr` by default:

    all.equal( coordinates(X, basis = 'ilr'),
               H1.ilr )
    #> [1] TRUE

## Other ilr-coordinates: Principal Components and Principal balances

Other easily accessible coordinates are the Principal Component (PC) coordinates. PC coordinates define the first coordinate as the log-contrast with highest variance, the second the one independent from the first and with highest variance and so on:

    H2.ilr = coordinates(X, basis = 'pc')
    head(H2.ilr)
    #>          pc1         pc2       pc3
    #> 1 -0.6787536  0.35694598 0.4319368
    #> 2 -0.5581520  0.57775877 0.5396259
    #> 3 -0.7013616  0.25302877 0.3467523
    #> 4 -0.8973701  0.25915667 0.3125234
    #> 5 -0.5362270 -0.05527103 0.1901418
    #> 6 -0.2676101  0.32802497 0.3852126
    barplot(apply(H2.ilr, 2, var))

Note that the PC coordinates are independent:

    cov(H2.ilr)
    #>              pc1           pc2           pc3
    #> pc1 4.475083e-01  1.036012e-16  1.997487e-16
    #> pc2 1.036012e-16  3.650673e-02 -3.031068e-17
    #> pc3 1.997487e-16 -3.031068e-17  1.257989e-02

The Principal Balance coordinates are similar to PC coordinates but with the restriction that the log contrast are balances

    H3.ilr = coordinates(X, basis = 'pb')
    head(H3.ilr)
    #>          pb1         pb2         pb3
    #> 1 -0.7026704 -0.14697799 -0.50925247
    #> 2 -0.5801749 -0.22679692 -0.74060456
    #> 3 -0.7206583 -0.12358191 -0.37622854
    #> 4 -0.9052439 -0.18836356 -0.33935049
    #> 5 -0.5646882  0.05082486 -0.07274761
    #> 6 -0.2956308 -0.07090708 -0.48495254
    barplot(apply(H3.ilr, 2, var))

Moreover, they are not independent:

    cor(H3.ilr)
    #>            pb1       pb2        pb3
    #> pb1  1.0000000 0.6043786 -0.3197742
    #> pb2  0.6043786 1.0000000  0.1594538
    #> pb3 -0.3197742 0.1594538  1.0000000

Principal Balances are hard to compute when the number of components is very high. `coda.base` allows to build PB approximations using different algorithms.

    X100 = exp(matrix(rnorm(1000*100), ncol = 100))

- Hierarchical clustering based algorithm.

    PB1.ward = pb_basis(X100, method = 'cluster')

- Constrained search algorithm

    PB1.constrained = pb_basis(X100, method = 'constrained')

We can compare they performance (variance explained by the first balance) with respect to the principal components.

    PC_approx = coordinates(X100, cbind(pc_basis(X100)[,1], PB1.ward[,1], PB1.constrained[,1]))
    names(PC_approx) = c('PC', 'Ward', 'Constrained')
    apply(PC_approx, 2, var)
    #>       h1       h2       h3
    #> 1.702951 1.418563 1.582705

Finally, `coda.base` allows to define the default CoDaPack basis which consists in defining well balanced balances, i.e. equal number of branches in each balance.

    H4.ilr = coordinates(X, basis = 'cdp')
    head(H4.ilr)
    #>        ilr1        ilr2        ilr3
    #> 1 0.7026704 -0.14697799 -0.50925247
    #> 2 0.5801749 -0.22679692 -0.74060456
    #> 3 0.7206583 -0.12358191 -0.37622854
    #> 4 0.9052439 -0.18836356 -0.33935049
    #> 5 0.5646882  0.05082486 -0.07274761
    #> 6 0.2956308 -0.07090708 -0.48495254

# Defining coordinates manually

## Defining coordinates with an specific basis

We can define the coordinates directly by providing the log-contrast matrix.

    B = matrix(c(-1,-1,2,0,
                 1,0,-0.5,-0.5,
                 -0.5,0.5,0,0), ncol = 3)
    H1.man = coordinates(X, basis = B)
    head(H1.man)
    #>          h1        h2          h3
    #> 1 -2.125532 0.5987412  0.10392914
    #> 2 -2.207723 0.4198053  0.16036964
    #> 3 -1.973384 0.6332727  0.08738560
    #> 4 -2.290402 0.7720507  0.13319315
    #> 5 -1.232257 0.6006268 -0.03593861
    #> 6 -1.277088 0.2454919  0.05013888

## Defining coordinates using balances

We can also define balances using formula `numerator~denominator`:

    B.man = sbp_basis(list(b1 = erc~jxcat,
                           b2 = psc~cs,
                           b3 = erc+jxcat~psc+cs),
                      data=X)
    H2.man = coordinates(X, basis = B.man)
    head(H2.man)
    #>            b1          b2        b3
    #> 1 -0.14697799 -0.50925247 0.7026704
    #> 2 -0.22679692 -0.74060456 0.5801749
    #> 3 -0.12358191 -0.37622854 0.7206583
    #> 4 -0.18836356 -0.33935049 0.9052439
    #> 5  0.05082486 -0.07274761 0.5646882
    #> 6 -0.07090708 -0.48495254 0.2956308

With `sbp_basis` we do not need to define neither a basis nor a system generator

    B = sbp_basis(list(b1 = erc+jxcat~psc+cs),
                  data=X)
    #> Warning in sbp_basis(list(b1 = erc + jxcat ~ psc + cs), data = X): Given
    #> partition is not a basis
    H3.man = coordinates(X, basis = B)
    head(H3.man)
    #>          b1
    #> 1 0.7026704
    #> 2 0.5801749
    #> 3 0.7206583
    #> 4 0.9052439
    #> 5 0.5646882
    #> 6 0.2956308

or

    B = sbp_basis(list(b1 = erc~jxcat+psc~cs,
                       b2 = jxcat~erc+psc+cs,
                       b3 = psc~erc+jxcat+cs,
                       b4 = cs~erc+jxcat+psc),
                  data=X)
    #> Warning in sbp_basis(list(b1 = erc ~ jxcat + psc ~ cs, b2 = jxcat ~ erc + :
    #> Given basis is not orthogonal
    H4.man = coordinates(X, basis = B)
    head(H4.man)
    #>            b1        b2         b3          b4
    #> 1 -0.01011597 0.5256940 -0.8214898  0.01011597
    #> 2 -0.26973693 0.5201431 -0.9396653  0.26973693
    #> 3  0.10888296 0.5169765 -0.7232616 -0.10888296
    #> 4  0.24556428 0.6764410 -0.7997213 -0.24556428
    #> 5  0.26662472 0.2845246 -0.3854211 -0.26662472
    #> 6 -0.22527958 0.2285779 -0.5666446  0.22527958

If interested, we can complete a sequential binary partition giving only some partitions

    B = sbp_basis(list(b1 = erc+jxcat~psc),
                  data=X, fill = TRUE)
    sign(B)
    #>    [,1] [,2] [,3]
    #> V1    1    1    1
    #> V2    1    1   -1
    #> V3   -1    1    0
    #> V4    0   -1    0

We can also define sequential binary partition using a matrix. By using a matrix we don’t need to include a dataset. The number of components is obtained with the number of rows and component names from row names (if available).

    P =  matrix(c(1, 1,-1,-1,
                  1,-1, 0, 0,
                  0, 0, 1,-1), ncol= 3)
    B = sbp_basis(P)
    H5.man = coordinates(X, basis = B)
    head(H5.man)
    #>          h1          h2          h3
    #> 1 0.7026704 -0.14697799 -0.50925247
    #> 2 0.5801749 -0.22679692 -0.74060456
    #> 3 0.7206583 -0.12358191 -0.37622854
    #> 4 0.9052439 -0.18836356 -0.33935049
    #> 5 0.5646882  0.05082486 -0.07274761
    #> 6 0.2956308 -0.07090708 -0.48495254